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9

Wormhole Solutions in Gauss-Bonnet-Born-Infeld Gravity

M. H. Dehghani1,2∗ and S. H. Hendi1,3†

1 Physics Department and Biruni Observatory,

College of Sciences, Shiraz University, Shiraz 71454, Iran

2 Research Institute for Astrophysics and Astronomy of Maragha (RIAAM), Maragha, Iran

3 Physics Department, College of Sciences, Yasouj University, Yasouj 75914, Iran

A new class of solutions which yields an (n+1)-dimensional spacetime with a longitudinal

nonlinear magnetic field is introduced. These spacetimes have no curvature singularity and

no horizon, and the magnetic field is non singular in the whole spacetime. They may be

interpreted as traversable wormholes which could be supported by matter not violating

the weak energy conditions. We generalize this class of solutions to the case of rotating

solutions and show that the rotating wormhole solutions have a net electric charge which

is proportional to the magnitude of the rotation parameter, while the static wormhole has

no net electric charge. Finally, we use the counterterm method and compute the conserved

quantities of these spacetimes.

I. INTRODUCTION

The intriguing possibility that our Universe is only a part of a higher dimensional spacetime

has raised a lot of interest in physics community recently. In string theory extra dimensions are

necessary since superstring theory requires a ten-dimensional spacetime to be consistent from the

quantum point of view. The effect of string theory on the left hand side of field equations of

gravity is usually investigated by means of a low energy effective action which describes gravity at

the classical level [1]. This effective action consists of the Einstein-Hilbert action plus curvature-

squared terms and higher powers as well, and in general give rise to fourth order field equations and

bring in ghosts. However, if the effective action contains the higher powers of curvature in particular

combinations, then only second order field equations are produced and consequently no ghosts arise

[2]. The effective action obtained by this argument is precisely of the form proposed by Lovelock

[3]. On the other hand, the dynamics of D-branes and some soliton solutions of supergravity is

governed by the Born-Infeld action [4], and therefore if one want to couple electromagnetic field

with gravity it is more suitable to put the energy momentum of Born-Infeld electromagnetic field

∗ mhd@shirazu.ac.ir† hendi@mail.yu.ac.ir

2

on the right hand side of the field equations. While the Lovelock gravity was proposed to have field

equations with at most second order derivatives of the metric [3], the nonlinear electrodynamics

proposed, by Born and Infeld, with the aim of obtaining a finite value for the self-energy of a

point-like charge [5]. The Lovelock gravity reduces to Einstein gravity in four dimensions and also

in the weak field limit, while the Lagrangian of the Born-Infeld (BI) electrodynamics reduces to

the Maxwell Lagrangian in the weak field limit. Considering the analogy between the Lovelock

and the Born-Infeld terms, which are on similar footing with regard to string corrections on the

gravity side and electrodynamic side, respectively, it is plausible to include both these corrections

simultaneously. Here we restrict ourself to the first three terms of Lovelock gravity. The first two

terms are the Einstein-Hilbert term with cosmological constant, while the third term is known as

the Gauss-Bonnet term. In recent decades, the exact solutions of Einstein-Gauss-Bonnet gravity

and their properties were studied. For example, static spherically symmetric black hole solutions of

the Gauss-Bonnet gravity were found in Ref. [6]. Black hole solutions with nontrivial topology in

this theory were also studied in Refs. [7, 8, 9]. NUT charged black hole solutions of Gauss-Bonnet

gravity and Gauss-Bonnet-Maxwell gravity were obtained in [10]. Also rotating black hole solutions

of Gauss-Bonnet-Maxwell and Born-Infeld-Gauss-Bonnet gravity have been studied in [11, 12].

Currently, there exist some activities in the field of wormhole physics following, particularly,

the seminal works of Morris, Thorne and Yurtsever [13]. Morris and Thorne assumed that their

traversable wormholes were time independent, non-rotating, and spherically symmetric bridges

between two universes. The manifold of interest is thus a static spherically symmetric spacetime

possessing two asymptotically flat regions. These kinds of wormholes could be threaded both by

quantum and classical matter fields that violate certain energy conditions at least at the throat

known as exotic matter. On general grounds, it has recently been shown that the amount of

exotic matter needed at the wormhole throat can be made arbitrarily small thereby facilitating

an easier construction of wormholes [14]. Lorentzian wormholes in spacetimes with more than

four dimensions were analyzed by several authors [15]. In particular, wormholes in Gauss-Bonnet

gravity were considered in Ref. [16]. In this paper we are looking for the (n + 1)-dimensional

horizonless solutions in Born-Infeld-Gauss-Bonnet gravity. The motivation for constructing these

kinds of solutions is that they may be interpreted as wormhole. Here, we will find these kinds of

horizonless solutions in the Born-Infeld-Gauss-Bonnet gravity, and use the counterterm method

to compute the conserved quantities of the system. The outline of our paper is as follows. In

Sec. II we give a brief review of the field equations of Born-Infeld-Gauss-Bonnet gravity, present

a new class of static wormhole solutions which produces longitudinal magnetic field, and then we

3

consider the properties of the solutions as well as the weak energy condition. In Sec. III we endow

these spacetime with global rotations and apply the counterterm method to compute the conserved

quantities of these solutions. Finally, we finish our paper with some closing remarks.

II. STATIC WORMHOLE SOLUTIONS

We begin this section with a brief review of Einstein-Gauss-Bonnet gravity in the presence of

nonlinear Born-Infeld electromagnetic field. The field equations of this theory in the presence of a

negative cosmological constant may be written as

Gµν −n(n− 1)

2l2gµν + αHµν =

1

2gµνL(F ) +

2FµλFνλ

√1 + F 2

2β2

, (1)

∂µ

√−gFµν√

1 + F 2

2β2

= 0, (2)

where Gµν is the Einstein tensor and Hµν is the divergence-free symmetric tensor

Hµν = 2R ρσλµ Rνρσλ − 4RρσRµρνσ + 2RRµν − 4RµλR

λν

−1

2gµν

(RκλρσR

κλρσ − 4RρσRρσ +R2

). (3)

In Eq. (1) β is the Born-Infeld parameter with dimension of mass, F 2 = FµνFµν where Fµν is

electromagnetic field tensor, and L(F ) is the Born-Infeld Lagrangian given as

L(F ) = 4β2

(

1 −√

1 +F 2

2β2

)

. (4)

In the limit β → ∞, L(F ) reduces to the standard Maxwell form L(F ) = −F 2, while L(F ) → 0 as

β → 0. Equation (1) does not contain the derivative of the curvatures, and therefore the derivatives

of the metric higher than two do not appear.

Here, we want to obtain the (n + 1)-dimensional solutions of Eqs. (1) and (2) which produce

longitudinal magnetic fields in the Euclidean submanifold spans by the xi coordinates (i = 1, ..., n−3). We assume that the metric has the following form:

ds2 = −r2

l2dt2 +

dr2

f(r)+ Γ2l2f(r)dψ2 + r2dφ2 +

r2

l2dX2, (5)

where dX2 =∑n−3

i=1 (dxi)2 and Γ is a constant will be fixed later. Note that the coordinates −∞ <

xi <∞ have the dimension of length, while the angular coordinates ψ and φ are dimensionless as

4

usual and range in [0, 2π]. The motivation for this metric gauge [gtt ∝ −r2 and (grr)−1

∝ gψψ]

instead of the usual Schwarzschild gauge [(grr)−1

∝ gtt] comes from the fact that we are looking

for a horizonless magnetic solution. The electromagnetic field equation (2) reduces to

β2l2Γ2[rF

′

ψr(r) + (n− 1)Fψr(r)]

+ (n− 1)F 3ψr(r) = 0, (6)

where the prime denotes a derivative with respect to the r coordinate. The solution of Eq. (6) can

be written as

Fψr =2Γ2ln−2q

rn−1√

1 − η, (7)

where q is an arbitrary constant and

η =4Γ2q2l2(n−3)

β2r2(n−1). (8)

Equation (7) shows that r should be greater than r01 = (2Γqln−3/β)1/(n−1) in order to have a real

nonlinear electromagnetic field and consequently a real spacetime. To find the function f(r), one

may use any components of Eq. (1). The simplest equation is the rr component of these equations

which can be written as

(n− 1){l2rn−4[2(n − 2)(n − 3)αf − r2]f′

+ (n− 2)l2rn−5[(n− 3)(n − 4)αf − r2]f

+nrn−1} + 4β2l2rn−1(1 −√

1 − η) = 0. (9)

The solution of Eq. (9), which also satisfies all the other components of the gravitational field

equations (1), can be written as

f(r) =r2

2 (n − 2) (n − 3)α

(1 −

√g(r)

), (10)

where

g(r) = 1 + 16(n − 3)αβ2η

n2F1

([1

2,n−2

2n−2

],

[3n−4

2n−2

], η

)

+4(n− 2)(n − 3)α

[4β2

(√1 − η − 1

)

n(n− 1)− 1

l2− 2ml

rn

], (11)

In Eq. (11) m is an integration constant which is related to geometrical mass of the spacetime and

2F1([a, b], [c], z) is the hypergeometric function which may be defined as [17]

2F1

([1

2,n−2

2n−2

],

[3n−4

2n−2

], bu2n−2

)=n− 2

un−2

∫un−3

√1 − bu2n−2

du.

5

One may note that the above asymptotically AdS solution reduces to Einstein-Born-Infeld solution

when α vanishes and reduces to the solution introduced in [18] as β goes to infinity. When m and

q are zero, the vacuum solution is

f(r) =r2

2(n − 2)(n − 3)α

(1 −

√1 − 4(n − 2)(n − 3)α

l2

). (12)

Equation (12) shows that for a positive value of α, this parameter should satisfies α ≤ l2/4(n −2)(n − 3). Also note that leff for the AdS solution of the theory is

leff = [2(n − 2)(n − 3)α]1/2

(

1 −√

1 − 4(n − 2)(n − 3)α

l2

)−1/2

, (13)

which reduces to l as α goes to zero.

To have a real spacetime, the function g(r) should be positive. This occurs provided the mass

parameter m ≤ m0, where m0 is the value of m calculated from the equation g(r = r01) = 0 given

as:

m0 =rn012l

{1

4(n − 2)(n − 3)α− 1

l2+

4β2

n(n− 2)2F1

([1

2,n−2

2n−2

],

[3n−4

2n−2

], 1

)− 4β2

n(n− 1)

}

In this case the metric function f(r) is real for r ≥ r0 = r01. For the case of m > m0, r should

be greater than r0 in order to have a real spacetime, where r0 is the largest real root of g(r) = 0.

Figure 1 shows the zeros of g(r) and 1 − η (r0 and r01 respectively) for m > m0.

–0.2

0

0.2

0.4

0.6

0.8

0.85 0.9 0.95 1 1.05 1.1 1.15r

FIG. 1: The functions g(r) (continuous-line) and 1 − η (dashed-line) versus r for n = 4, q = 0.3, m = 0.8,

l = Γ = β = 1, and α = 0.2 .

In order to study the general structure of this solution, we first look for curvature singularities.

It is easy to show that the Kretschmann invariant RµνλκRµνλκ diverges at r = r0, it is finite for

r > r0 and goes to zero as r → ∞. Therefore one might think that there is a curvature singularity

6

located at r = r0. Two cases happen. In the first case the function f(r)has no real root greater

than r0, and therefore we encounter with a naked singularity which we are not interested in it.

So we consider only the second case which the function has one or more real root(s) larger than

r0. In this case the function f(r) is negative for r < r+, and positive for r > r+ where r+ is the

largest real root of f(r) = 0. Indeed, grr and gψψ are related by f(r) = g−1rr = Γ−2l−2gψψ, and

therefore when grr becomes negative (which occurs for r0 < r < r+) so does gψψ. This leads to

an apparent change of signature of the metric from (n− 1)+ to (n− 2)+, and therefore indicates

that r should be greater than r+. Thus the coordinate r assumes the value r+ ≤ r < ∞. The

function f(r) given in Eq. (10) is positive in the whole spacetime and is zero at r = r+. The

Kretschmann scalar is a linear combination of the square of f ′′, f ′/r and f/r2. Since these terms

do not diverge in the range r+ ≤ r <∞, one finds that the Kretschmann scalar is finite. Also one

can show that other curvature invariants (such as Ricci scalar, Ricci square, Weyl square and so

on) are functions of f ′′, f ′/r and f/r2 and therefore the spacetime has no curvature singularity in

the range r+ ≤ r < ∞. In order to avoid conic singularity at r = r+ in the (r − ψ)-section, one

may fix the factor Γ = 1/[lf ′(r+)] in the metric.

Now, we investigate the wormhole interpretation of the above solution. It satisfies the so-called

flare-out condition when r = r+. This can be seen by embedding the 2-surface of constant t, ψ

and xi’s with the metric

ds2 =dr2

f(r)+ r2dφ2, (14)

in an (unphysical) three-dimensional Euclidean flat space, which has the metric

ds2 = dr2 + r2dφ2 + dz2 (15)

in cylindrical coordinates. The surface described by the function z = z(r) satisfies

dr

dz=

√f(r)

1 − f(r)= 0, (16)

d2r

dz2=

f ′

2 [1 − f ]2> 0, (17)

which show that it has the characteristic shape of a wormhole, as illustrated in Figs. 1 and 2 of

Ref. [13]. Next we consider the Weak Energy Condition (WEC) for the above solution. Using the

orthonormal contravariant (hatted) basis vectors

ebt =l

r

∂

∂t, ebr = f1/2 ∂

∂r, e bψ

=1

Γlf1/2

∂

∂ψ, ebφ

= r−1 ∂

∂φ, e bxi

=l

r

∂

∂xi,

7

the mathematics and physical interpretations become simplified. It is a matter of straight forward

calculations to show that the components of stress-energy tensor are

Tbtbt

= −Tbφbφ

= −Tbibi

=β2

4π

[

1 +

(FψrΓlβ

)2]1/2

− 1

,

Tbrbr

= Tbψ bψ

=β2

4π

1 −[1 +

(FψrΓlβ

)2]−1/2

, (18)

which satisfy the weak energy conditions:

Tbtbt≥ 0, T

btbt+ T

bibi= T

btbt+ T

bφbφ≥ 0, T

btbt+ T

brbr= T

btbt+ T

bψ bψ≥ 0. (19)

III. ROTATING WORMHOLE SOLUTIONS

First, we want to endow our spacetime solution (5) with a global rotation. In order to add

angular momentum to the spacetime, we perform the following rotation boost in the t− ψ plane

t 7→ Ξt− aψ, ψ 7→ Ξψ − a

l2t, (20)

where a is a rotation parameter and Ξ =√

1 + a2/l2. Substituting Eq. (20) into Eq. (5) we obtain

ds2 = −r2

l2(Ξdt− adψ)2 +

dr2

f(r)+ Γ2l2f(r)

( al2dt− Ξdψ

)2+ r2dφ2 +

r2

l2dX2, (21)

where f(r) is the same as f(r) given in Eq. (10). The non-vanishing components of electromagnetic

field tensor are now given by

Ftr =a

Ξl2Frψ =

2Γ2ln−4qa

rn−1√

1 − η. (22)

Because of the periodic nature of ψ, the transformation (20) is not a proper coordinate transfor-

mation on the entire manifold. Therefore, the metrics (5) and (21) can be locally mapped into

each other but not globally, and so they are distinct [19]. Note that this spacetime has no horizon

and curvature singularity. One should note that these solutions are different from those discussed

in [11], which were electrically charged rotating black brane solutions in Gauss-Bonnet gravity.

The electric solutions have black holes, while the magnetic solution interpret as wormhole. It

is worthwhile to mention that this solution reduces to the solution of Einstein-Maxwell equation

introduced in [18] as α goes to zero.

Second, we generalize the above solution to the case of rotating solution with more rotation

parameters. The rotation group in n + 1 dimensions is SO(n) and therefore the number of inde-

pendent rotation parameters is [n/2], where [x] is the integer part of x. The generalized solution

8

with k ≤ [n/2] rotation parameters can be written as

ds2 = −r2

l2

(

Ξdt−k∑

i=1

aidψi

)2

+ Γ2f(r)

(√

Ξ2 − 1dt− Ξ√Ξ2 − 1

k∑

i=1

aidψi

)2

+dr2

f(r)+

r2

l2(Ξ2 − 1)

k∑

i<j

(aidψj − ajdψi)2 + r2dφ2 +

r2

l2dX2, (23)

where Ξ =√

1 +∑k

i a2i /l

2, dX2 is the Euclidean metric on the (n−k−2)-dimensional submanifold

and f(r) is the same as f(r) given in Eq. (10). The non-vanishing components of electromagnetic

field tensor are

Ftr =(Ξ2 − 1)

ΞaiFrψi =

2qln−3Γ2√

Ξ2 − 1

rn−1√

1 − η. (24)

A. Conserved Quantities

In general the conserved quantities are divergent when evaluated on the solutions. A systematic

method of dealing with this divergence for asymptotically AdS solutions of Einstein gravity is

through the use of the counterterms method inspired by the anti-de Sitter conformal field theory

(AdS/CFT) correspondence [20]. For asymptotically AdS solutions of Lovelock gravity with flat

boundary, R̂abcd(γ) = 0, the finite energy momentum tensor is [21, 22]

T ab =1

8π{(Kab −Kγab) + 2α(3Jab − Jγab) −

(n− 1

L

)γab}, (25)

where L is

L =3√χ(1 −√

1 − χ)3/2

√8[1 −

(1 + χ

2

)√1 − χ

] l, (26)

and χ = 4(n− 2)(n− 3)α/l2. In Eq. (25), Kab is the extrinsic curvature of the boundary, K is its

trace, γab is the induced metric of the boundary, and J is trace of Jab

Jab =1

3(KcdK

cdKab + 2KKacKcb − 2KacK

cdKdb −K2Kab). (27)

One may note that when α goes to zero, the finite stress-energy tensor (25) reduces to that of

asymptotically AdS solutions of Einstein gravity with flat boundary.

To compute the conserved charges of the spacetime, we choose a spacelike surface B in ∂Mwith metric σij , and write the boundary metric in ADM (Arnowitt-Deser-Misner) form:

γabdxadxa = −N2dt2 + σij

(dφi + V idt

) (dφj + V jdt

), (28)

9

where the coordinates φi are the angular variables parameterizing the hypersurface of constant r

around the origin, and N and V i are the lapse and shift functions respectively. When there is a

Killing vector field ξ on the boundary, then the quasilocal conserved quantities associated with the

stress tensors of Eq. (25) can be written as

Q(ξ) =

∫

B

dn−1ϕ√σTabn

aξb, (29)

where σ is the determinant of the metric σij , and na is the timelike unit normal vector to the

boundary B. In the context of counterterm method, the limit in which the boundary B becomes

infinite (B∞) is taken, and the counterterm prescription ensures that the action and conserved

charges are finite. No embedding of the surface B in to a reference of spacetime is required and

the quantities which are computed are intrinsic to the spacetimes.

For our case, the magnetic solutions of Gauss-Bonnet gravity, the first Killing vector is ξ = ∂/∂t,

therefore its associated conserved charge is the total mass of the wormhole per unit volume Vn−k−2,

given by

M =

∫

B

dn−1x√σTabn

aξb =(2π)k

4

[n(Ξ2 − 1) + 1

]Γm. (30)

For the rotating solutions, the conserved quantities associated with the rotational Killing symme-

tries generated by ζi = ∂/∂φi are the components of angular momentum per unit volume Vn−k−2

calculated as

Ji =

∫

B

dn−1x√σTabn

aζbi =(2π)k

4ΓnΞmai. (31)

Next, we calculate the electric charge of the solutions. To determine the electric field we should

consider the projections of the electromagnetic field tensor on special hypersurfaces. The normal

to such hypersurfaces for the spacetimes with a longitudinal magnetic field is

u0 =1

N, ur = 0, ui = −N

i

N,

and the electric field is Eµ = gµρFρνuν . Then the electric charge per unit volume Vn−k−2 can be

found by calculating the flux of the electromagnetic field at infinity, yielding

Q =(2π)k

2Γq√

Ξ2 − 1. (32)

Note that the electric charge is proportional to the magnitude of rotation parameters and is zero

for the static solutions.

10

IV. CLOSING REMARKS

Considering both the nonlinear invariant terms constructed by electromagnetic field ten-

sor (FµνFµν) and quadratic invariant terms constructed by Riemann tensor (R2, RµνR

µν and

RµνρσRµνρσ) in action, we have obtained a new class of rotating spacetimes in various dimensions,

with negative cosmological constant. These solutions are asymptotically anti-de Sitter and reduce

to the solutions of Gauss-Bonnet-Maxwell gravity when β → ∞ and reduce to those of Einstein-

Born-Infeld gravity as α→ 0. This class of solutions yields an (n+ 1)-dimensional spacetime with

a longitudinal nonlinear and nonsingular magnetic field (the only nonzero component of the vector

potential is Aφ(r)) generated by a static magnetic source. We have found that these solutions have

no curvature singularity and no horizons and may be interpreted as a traversable wormhole near

r = r+. Also, we found that the weak energy condition is not violated at the throat, which shows

there is no exotic matter near the throat.

Also we have generalized these solutions to the case of rotating spacetimes with a longitudinal

magnetic field. For the rotating wormhole, when a rotational parameter is nonzero, the wormhole

has a net electric charge density which is proportional to the magnitude of the rotational parameter

given by√

Ξ2 − 1. For static case, the electric field vanishes, and therefore the wormhole has no

net electric charge. Finally, we applied the counterterm method and calculated the conserved

quantities of the solutions. We have found that the conserved quantities do not depend on the

Born-Infeld parameter β.

Acknowledgments

This work has been supported by Research Institute for Astrophysics and Astronomy of

Maragha.

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